In 1927, Dirac proposed that quantum mechanics may be elevated to a relativistic theory by quantizing a spinor field [1].
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Theory
Kinematics
During this calculation, we shall make use of two different inertial reference frames to obtain our final result. Mandelstam variables are constructed from the incoming and outgoing momenta of the interaction. \[
\begin{array}{lllllllll}
s &=& (p+k)^{2} &=& p^{2}+k^{2}+2 p \cdot k &=& m^{2}+2 p \cdot k &=& m^{2}+2 p^{\prime} \cdot k^{\prime} \\
t &=& \left(p^{\prime}-p\right)^{2} &=& p^{\prime 2}+p^{2}-2 p \cdot p^{\prime} &=& 2 m^{2}-2 p \cdot p^{\prime} &=& -2 k \cdot k^{\prime} \\
u &=& \left(k^{\prime}-p\right)^{2} &=& k^{\prime 2}+p^{2}-2 k^{\prime} \cdot p &=& m^{2}-2 k^{\prime} \cdot p &=& m^{2}-2 k \cdot p^{\prime}
\end{array}
\] Casting transition amplitudes in terms of these manifestly Lorentz invariant quantities allows us to jump between frames with ease.
Centre-of-mass frame
Inetial frame in which sum of spatial momenta is zero.
Centre of mass frame diagram
Lab frame
Inertial frame in which the electron is at rest. This will also be the assumed rest frame for our particle detectors, hence lab frame.
By applying the Feynman rules to these diagrams and grouping terms, we obtain \[
i \mathcal{M}=i e^{2}
\epsilon_{\mu \lambda^\prime}^{\ast}
\left(k^{\prime}\right)
\epsilon_{\nu \lambda}\left(k\right)
\bar{u}^{s^{\prime}}\left(p^{\prime}\right)
\left(
\dfrac{% numerator
\gamma^{\mu}(\not{p}+k+m) \gamma^{\nu}}
{(p+k)^{2}-m^{2}}% denominator
+ \dfrac{% numerator
\gamma^{\nu}
\left(\not{p}-k^{\prime}+m\right)
\gamma^{\mu}}
{\left(p-k^{\prime}\right)^{2}-m^{2}}% denominator
\right)
u^{s}(p)
\]
This unwieldy expression can be simplified a little by expanding the binomials in the denominator, and observing for the numerator \[
\begin{aligned}
(\not{p}+m) \gamma^{\nu} u^{s}(p) &=\left(2 p^{\nu}-\gamma^{\nu} \not{p}+\gamma^{\nu} m\right) u^{s}(p) \\
&=2 p^{\nu} u^{s}(p)-\gamma^{\nu}\underbrace{(\not{p}-m) u^{s}(p)}_{
\text{Dirac equation} \implies 0
} \\
&=2_{} p^{\nu} u^{s}(p)
\end{aligned}
\]
